Less than or equal sign

You have a sound understanding of the matter:

It is true that $a\lt b \implies a\leq b$, since $$a \leq b \iff (a\lt b\;\text{or}\;a = b)$$

If you need to prove $a \leq b$, then it suffices to prove either $a < b$ or $a = b$.

On the other hand, if you need (or want) to prove $a$ is strictly less than $b$, then it is necessary to prove $a < b$.

There can be, indeed, some loss of information, just as we lose information when, while knowing $x = 3$, we assert only the (true) claim that $x \leq 3$.

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But there are examples where we can make a claim for a wider range of values by using the inclusive $\leq$.

I don't know if this example will be of any help, but if you are trying to prove, by induction, that $n^2 \leq 2^n$ for $n>3$, in your inductive proof, we first have the base case $n = 4$, where the inequality holds precisely because equality holds: $4^2 = 2^4$, hence $4^2 \leq 2^4$ is true.

But in the inductive step, you might use the following chain: $$(n+1)^2 = n^2 + 2n + 1 \leq 2^n + \underbrace{2n + 1}_{\large <\; 2^n,\,n\geq 3} \lt 2^n + 2^n = 2\cdot 2^n = 2^{n+1}$$

So in the inductive step, even though we find we have a strict inequality involved, the point of the induction is to prove $n^2 \leq 2^n$, for $n \gt 3$ and because the base case is true (because equality holds), we can actually say more by affirming the NON-strict inequality (since the range of $n$ for which the proposition is true is greater with $\leq$ than with $\lt$.)

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In the end, the amount of information conveyed by the choice of using $\lt$ vs. $\leq$ depends, as do many choices, on context.

Yes. Since $a\leq b \Leftrightarrow (a<b \vee a=b)$, $a<b$ implies $a\leq b$.

It is correct to say that if $a < b$, then $a \le b$. And yes, I completely agree with you: if explicitly being shown $a < b$, there is a certain "loss" of information when we write $a \le b$ instead of being concrete.

"$\le$" is, however, very useful in real-life, practical mathematical problems such as optimization when you want to say that one quantity does not exceed another.